The characteristic polynomial of an algebra and representations

We pose two new linear preserver problems. First, given a field k and a finite dimensional k-algebra B we show that the only linear maps phi: k(xd) -> B which preserve the unit and the n-th roots of unity (for some n > 2 coprime to char(k)) are the algebra homomorphisms. Second, we consider linear maps phi: A -> B between finite dimensional k-algebras which preserve the Cayley Hamilton relations. We show that if preservation of the Cayley Hamilton property is understood in a certain non-commutative sense, and A = k(Xd), then the only such linear mappings are the algebra homomorphisms. (C) 2017 Elsevier Inc. All rights reserved.